Abstract
In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of GelfandKirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following theorem. Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and let A be any connected graded noetherian maximal order containing R, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Secondly, let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules).
Original language  English 

Article number  107151 
Number of pages  42 
Journal  Advances in Mathematics 
Volume  369 
Early online date  27 Apr 2020 
DOIs  
Publication status  Published  5 Aug 2020 
Keywords
 math.RA
 math.AG
 math.QA
 Primary: 14A22, 16P40, 16S38, 16W50, Secondary: 14H52, 14E30
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Sue Sierra
 School of Mathematics  Personal Chair of Noncommutative Algebra
Person: Academic: Research Active